To understand the mechanism of maintenance of genetic variability and evolution of populations, the following four problems will be studied. (1) Genetic differentiation of populations. Mathematical theories of the gene differentiation between populations under the effect of natural selection and the distribution of genetic distance among loci will be developed. Using the theories developed and gene frequency data compiled from the literature, we intend to study the pattern of human race evolution. Mathematical studies will also be made on the genetic differentiation of quantitative characters under the joint effect of mutation, selection, and random genetic drift as well as on the mechanism of speciation. (2) Mechanism of maintenance of polymorphic genes. We plan to study the relative importance of deterministic an stochastic factors in the maintenance of protein polymorphism. We are particularly interested in testing the null hypothesis of neutral mutations by means of statistical analyses of gene frequency data. We shall also investigate other hypotheses that are capable of explaining polymorphism data. (3) Maintenance of deleterious genes in finite populations. The growth and the distribution of the number of deleterious genes in human populations will be studied by using stochastic models. The transient distribution of deleterious gene frequencies in finite populations will also be studied to clarify the population dynamics of recessive deleterious mutations. Furthermore, both empirical and theoretical studies will be made about the racial variation in the frequencies of genetic diseases and the factors responsible for this variation. (4) Utility of linked marker genes for genetic counseling. The theoretical basis of using marker genes for genetic counseling will be clarified by considering the gene frequencies and dominance relationship at the marker loci and the recombination value between the disease and maker loci.